Tagged Questions

5
votes
3answers
58 views

Question Regarding the Axiom of Extensionality

Jech's text on Set Theory states the following: If X and Y have the same elements, then X = Y : ∀u(u ∈ X ↔ u ∈ Y ) → X = Y. The converse, namely, if X = Y then u ∈ X ↔ u ∈ Y, is an axiom ...
8
votes
1answer
71 views

What is the smallest fragment of ZFC that has the same consistency strength as ZFC?

The question in the title is undoubtedly nonsensical, but I am not sure how to state this question properly. Perhaps some examples will help me explain it. Thanks to Godel and Cohen, we know that ...
-2
votes
0answers
36 views

Empty Set Axiom + Extensionality Axiom [closed]

I am trying to prove the following two sentences are equivalent: 1) Empty Set Axiom + Extensionality Axiom 2) $(\exists A) (\forall x) \neg(x\in A)$ + Extensionality Axiom Obviously, as one can see ...
4
votes
2answers
45 views

ZF Extensionality axiom

To familiarize myself with axiomatic set theory, I am reading Kenneth Kunen's The Foundations of Mathematics that presents ZF set theory. I haven't gotten really far since I am stuck at the axiom of ...
3
votes
2answers
70 views

Axiom of infinity exceptions?

It is my understanding that the axiom of infinity, or axioms that depend on the axiom of infinity, are the only axioms that are not representable in finite First-Order Predicate Logic. Is this ...
3
votes
2answers
28 views

Schema of separation and set of all sets

The schema of separation states (this is probably simplified) states that if $P$ is a property and $X$ is a set, then there exists a set $Y = \{ x \in X : P(x)\}$. The notes I'm reading say that from ...
4
votes
3answers
133 views

A basic doubt on axiom of foundation of Zermelo-Fraenkel set theory

I have read in a book that the "axiom of foundation prevents anomalies such as a set being an element of itself". Now, axiom of foundation says that there exist an element in every set which is ...
1
vote
0answers
60 views

Is a “model” only a proper model if everything in it's definition is also explicitly constructed?

Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the ...
2
votes
1answer
50 views

axiom of foundation of Zermelo–Fraenkel set theory

I have found two different statements on axiom of foundation of Zermelo–Fraenkel set theory in two different books as: 1) every nonempty set contains an element that is not an element of any other ...
3
votes
0answers
35 views

Have people studied weakened forms of Tarski's axiom?

My intuition about Grothendieck universes says the following. Axiom of empty set $\leftrightarrow$ There exists a Grothendieck universe. Axiom of infinity $\leftrightarrow$ There exist at least two ...
5
votes
3answers
129 views

The axiom of infinity for Zermelo–Fraenkel set theory

The axiom of infinity for Zermelo–Fraenkel set theory is stated as follows in the wikipedia page: Let $S(w)$ abbreviate $ w \cup \{w\} $, where $ w $ is some set (We can see that $\{w\}$ is a ...
2
votes
1answer
46 views

Formulation of the Separation Axiom in Set Theory

I have a question regarding this formulation of the Separation Axiom: I quote from the Book "Handbook of Mathematical Logic", the chapter from which this is taken could also be found here. Now we ...
4
votes
1answer
115 views

Is ZFC without Axiom of Infinity consistent?

The incompleteness theorem states that one cannot prove whether ZF or ZFC is consistent, but what about ZFC withouth Axiom of infinity? (Assuming the empty set exists) Furthermore, let $M$ be a ...
16
votes
5answers
400 views

Why hasn't GCH become a standard axiom of ZFC?

I've never seen a text that includes GCH in the ZFC axioms. I presume this means that GCH has not achieved widespread acceptance. This seems surprising to me, given that: The cardinal numbers ...
2
votes
2answers
99 views

Would it be possible to concoct a “harmful” axiom?

Suppose I run an automated theorem prover. It begins with the axioms of ZFC, and using a random number generator, it proves more theorems, and it runs for two days. At the end of the second day, it ...
1
vote
1answer
60 views

What makes Tarski Grothendieck set theory non-empty?

I'm fighting with Grothendieck set theory for some time now. This is the framework for the automated proof checking system of Mizar and hence there is a formalized version of the axioms here too, and ...
4
votes
2answers
190 views

How does (ZFC-Infinity+“There is no infinite set”) compare with PA?

How does (ZFC-Infinity+"There is no infinite set") compare with (first order) PA? Intuitively, neither theory should be more powerful than the other.
2
votes
2answers
68 views

Why does any transitive model satisfy extensionality?

I see it stated as something very clear, but i can't figure it out. i found a proof in Jech(old version) which goes through the concept of restricted formulas, which i don't quite understand. (i'm not ...
2
votes
1answer
142 views

Explain Zermelo–Fraenkel set theory in layman terms

What does Zermelo–Fraenkel set theory mean? According to Wikipedia, Zermelo–Fraenkel set theory is a set theory that is proposed to overcome issues in naive set theory. I really appreciate if ...
2
votes
2answers
115 views

Forcing Question about a sequence of functions from $\aleph_{0}$ into $2 = \{0, 1\}$

I'm trying to go through Timothy Chow's A Beginner's Guide to Forcing (arxiv pdf). My particular question is about the first paragraph of Section $5$ (on p. $7$ of the above pdf). First, my vague ...
5
votes
3answers
176 views

The existence of the empty set is an axiom of ZFC or not?

I found in the Wolfram MathWorld page of the Axiom of the Empty Set that this is one of the Zermelo-Fraenkel Axioms, however on the page about these ZFC Axioms I read that it is an axiom that can be ...
3
votes
3answers
203 views

What properties are allowed in comprehension axiom of ZFC?

I am trying to understand the axioms of ZFC. The axiom schema of specification or the comprehension axioms says: If A is a set and $\phi(x)$ formalizes a property of sets then there exists a set C ...
1
vote
1answer
68 views

What will happen when Ax.Inf is replaced to its negation in ZFC? [duplicate]

Possible Duplicate: What are the consequences if Axiom of Infinity is negated? In ZFC, if we replace Ax.Inf to such a statement that every set is finite, then does this theory satisfiable? ...
8
votes
2answers
318 views

Where is axiom of regularity actually used?

Where is axiom of regularity actually used? Why is it important? Are there some proofs, which are substantially simpler thanks to this axiom? This question was to some extent provoked by Dan ...
1
vote
1answer
120 views

What should I be able to do with this chapter on Axiomatic Set Theory in order to check if I've learned it decently? [closed]

I've just read a chapter on axiomatic set theory, from Comprehensive Mathematics for Computer Scientists 1. It comes with basic notation on sets and some axioms: Axiom 1 (Axiom of Empty Set) Axiom 2 ...
3
votes
1answer
201 views

Are there ways to describe the Martin Axiom intuitively?

I'm looking for some way of understanding the Martin Axiom for some $\kappa$ in an intuitive way. I know that the motivation is the Baire Category Theorem, however is there any other way of thinking ...
5
votes
2answers
250 views

What interpretations exists for the singleton of the singleton of the empty set $\{\{\varnothing\}\}$ and its singleton $\{\{\{\varnothing\}\}\}$?

What I know I know from Peano's axioms that the empty set is equivalent to the natural number $0$ and that the singleton of the empty set is equivalent to the natural number $1$. ( ...
2
votes
1answer
176 views

How do I prove the existence of infinite union in ZFC?

Given an infinite set of sets A - how can I prove in ZFC that the union of all the elements of A exists?
3
votes
1answer
98 views

Does (Infer $\phi$ from $\psi$) imply (Infer $\phi^L$ from $\psi^L$)?

I am studying set theory on my own on Drake's famous book and I'm stuck on the (finitary) prove of the relative consistency of the Axiom of Choice. Is it true that a if we were able to infer $\xi$ ...
2
votes
2answers
119 views

Am I allowed to realize one object twice within one set-theory?

Say I consider a set theory with the Axioms of Extensionality and the Axiom of Pairing. As I understand it, stating the axiom allows me to make a definition like $$(a,b):=\{\{a\},\{a,b\}\}$$ and ...
2
votes
3answers
240 views

Axiom of subsets and Russell's paradox

Russell, with his paradox, proves that the set $\{x:x\notin x\}$ of all sets that are not members of themselves doesn't exist. So, he demonstrates that the set $\{x:p(x)\}$ doesn't exist necessary (it ...
2
votes
1answer
164 views

Proving the pairing axiom from the rest of ZF

In ZF, the pairing axiom states that for every $x,y$ there exists the set $\{x, y\}$. Wikipedia also tells us we can dispense this axiom: This axiom is part of Z, but is redundant in ZF because it ...
1
vote
1answer
171 views

Do the proofes in set theory rely on the semantics of the formulas used in the axioms?

Motivation: The Axiom of separation $$\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \phi(x, w_1, \ldots, w_n, A) ] )$$ is used to ...
3
votes
3answers
149 views

An elementary question regarding the uniqueness of a set, viewed with different cardinality

Does the cardinality of sets, like for example the real numbers, depend on the fundamental axioms one is working with? If so, what does it mean to speak of such a set if it is not really one single ...
4
votes
1answer
232 views

Need help understanding Axiom of Extensionality

I'm attempting to learn Set Theory and I'm currently working through Halmos' Naive Set Theory. I will say that I completely understand the essence of the Axiom of Extensionality. However, where I'm ...
12
votes
1answer
524 views

Proofs given in undergrad degree that need Continuum hypothesis?

Or alternative you need to assume CH is false. I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non ...
5
votes
1answer
275 views

Can the power set be axiomatised?

I want to consider many-sorted first order logic with distinguished sorts $U$ and $P$. Can I state a (finite?) set of first order formulae such that any model $M = (D^U, D^P, I)$ interprets the sort ...
3
votes
1answer
221 views

Can it be shown that ZFC has statements which cannot be proven to be independent, but are?

I am familiar with the concept that a statement can be proven indepent such as in the case of the continuum hypothesis where both ZFC+CH and ZFC+(CH is false) are both proven consistent, but I would ...
20
votes
6answers
1k views

What are natural numbers?

What are the natural numbers? Is it a valid question at all? My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational ...
7
votes
2answers
347 views

Relationship between Category theory and Axiomatic set theory

I've recently started learning Category theory- and I have a pondering- wondering if anyone can help. Is it possible for two categories to satisfy two different set-axiom system. Namely- is it ...
6
votes
9answers
527 views

Motivating implications of the axiom of choice?

What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some ...
5
votes
2answers
483 views

How can I write the Axiom of Specification as a sentence?

I began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification". To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly ...
33
votes
4answers
4k views

Can you explain the “Axiom of choice” in simple terms?

As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. I went to Wikipedia to see what the Axiom of ...